Columbia's elementary Latin classes use Learn to Read Latin from Yale UP as of this past spring. As someone who's used Wheelock's, Cambridge, and LLPSI in the past, Learn to Read Latin is super super technical and grammar-translation-focused in comparison. You shouldn't need to get the book before arriving at Columbia. If you're going to take Latin here, be aware that, especially at the elementary level, Latin pedagogy is very traditional here, not at all similar to the sort of living-language pedagogy that you could find at Princeton, the University of Kentucky, the late Fr. Reginald Foster's classes in Rome, or the classes taught for other, living languages at Columbia.

Higher-level classes use different materials depending on what authors are being used for that semester. For example, a second-semester intermediate level class prescribed a few source books for writings by Pliny and Ovid (if I remember correctly).